Abstract
This paper introduces the theory of bi-decomposition of Boolean functions. This approach optimally exploits functional properties of a Boolean function in order to find an associated multilevel circuit representation having a very short delay by using simple two input gates. The machine learning process is based on the Boolean Differential Calculus and is focused on the aim of detecting the profitable functional properties available for the Boolean function.
For clear understanding the bi-decomposition of completely specified Boolean functions is introduced first. Significantly better chance of success are given for bi-decomposition of incompletely specified Boolean functions, discussed secondly. The inclusion of the weak bidecomposition allows to prove the the completeness of the suggested decomposition method. The basic task for machine learning consists of determining the decomposition type and dedicated sets of variables. Lean on this knowledge a complete recursive design algorithm is suggested.
Experimental results over MCNC benchmarks show that the bi-decomposition outperforms SIS and other BDD-based decomposition methods in terms of area and delay of the resulting circuits with comparable CPU time.
By switching from the ON-set/OFF-set model of Boolean function lattices to their upperand lower-bound model a new view to the bi-decomposition arises. This new form of the bi-decomposition theory makes a comprehensible generalization of the bi-decomposition to multivalued function possible.
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Abbreviations
- BDC:
-
Boolean Differential Calculus
- BDD:
-
Binary Decision Diagram
- ISF:
-
Incompletely Specified Function.
References
Akers, S. B. (1959). On a Theory of Boolean Functions. J. Soc. Ind. Appl. Math. 7(4).
Ashenhurst, R. (1957). The Decomposition of Switching Functions. In International Symposium on the Theory of Switching Functions, 74–116.
Bochmann, D., Dresig, F. & Steinbach, B. (1991). A New Decomposition Method for Multilevel Circuit Design. In European Conference on Design Automation, 374–377. Amsterdam, Holland.
Bochmann, D. & Posthoff, C. (1981). Binäre dynamische Systeme. München, Germany: Oldenbourg Verlag.
Bochmann, D. & Steinbach, B. (1991). Logikentwurf mit XBOOLE. Berlin, Germany: Verlag Technik.
Böhlau, P. (1987). Eine Dekompositionsstrategie für den Logikentwurf auf der Basis funktions-typischer Eigenschaften. Dissertation thesis, Technical University Karl-MarxStadt, Germany.
Curtis, H. (1962). A New Approach to the Design of Switching Circuits. Princeton, USA: Van Nostrand.
Dresig, E (1992). Gruppierung — Theorie und Anwendung in der Logiksynthese. Düsseldorf, Germany: VDI-Verlag.
Lang, C. (2003). Bi-Decomposition of Function Sets Using Multi-Valued Logic. Dissertation thesis, Freiberg University of Mining and Technology, Germany.
Le, T. Q. (1989). Testbarkeit kombinatorischer Schaltungen — Theorie und Entwurf. Dissertation thesis, Technical University Karl-Marx-Stadt, Germany.
Mishchenko, A., Steinbach, B. & Perkowski, M. (2001). An Algorithm for Bi-Decomposition of Logic Functions. In 38th Design Automation Conference, 18–22. Las Vegas, USA.
Sasao, T. & Butler, J. (1997). On Bi-Decompositions of Logic Functions. In International Workshop on Logic Synthesis, 18–21. Lake Tahoe, USA
Sentovich, e. a. (1992). SIS: A System for Sequential Circuit Synthesis. University of California, Berkeley, California, USA. Technical Report UCB/ERI, M92/41, ERL, Dept. of EEC S .
Somenzi, F. (2001). Binary Decition Diagram (BDD) Package: CUDD v. 2.3.1. University of Colorado at Boulder. URL http://vlsi.colorado.edu/—fabio/CUDD/cuddIntro.html.
Steinbach, B. & Hesse, K. (1996). Design of Large Digital Circuits Utilizing Functional and Structural Properties. In 2nd Workshop on Boolean Problems, 23–30. Freiberg, Germany.
Steinbach, B. & Le, T. Q. (1990). Entwurf testbarer Schaltnetzwerke. Wissenschaftliche Schriftenreihe 12/1990, Technical University Chemnitz, Germany.
Steinbach, B., Schuhmann, E & Stöckert, M. (1993). Functional Decomposition of Speed Optimized Circuits. In Auvergne, D. & Hartenstein, R. (eds.) Power and Timing Modelling for Performance of Integrated Circuits, 65–77. Bruchsal, Germany: IT Press.
Steinbach, B. & Stöckert, M. (1994). Design of Fully Testable Circuits by Functional Decomposition and Implicit Test Pattern Generation. In 12th IEEE VLSI Test Symposium, 22–27.
Steinbach, B. & Wereszczynski, A. (1995). Synthesis of Multi-Level Circuits Using EXORGates. In IFIP WG 10.5 Workshop on Applications of the Reed-Muller Expansion, 161–168. Chiba, Japan.
Steinbach, B. & Zakrevski, A. (1998). Three Models and Some Theorems on Decomposition of Boolean Functions. In 3rd International Workshop on Boolean Problems, 11–18. Freiberg, Germany.
Steinbach, B. & Zhang, Z. (1997). Synthesis for Full Testability of Partitioned Combinational Circuits Using Boolean Differential Calculus. In 6th International Workshop on Logic and Synthesis, 1–4. Granlibakken, USA.
Yang, M. C. (2000). BDD-Based Logic Optimization System. Technical Report CSE-00–1.
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Steinbach, B., Lang, C. (2004). Exploiting Functional Properties of Boolean Functions for Optimal Multi-Level Design by Bi-Decomposition. In: Artificial Intelligence in Logic Design. Artificial Intelligence in Logic Design, vol 766. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-2075-9_6
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DOI: https://doi.org/10.1007/978-1-4020-2075-9_6
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